Collections, like people, have personalities. The personalities of good or very good collections are recognizable while remaining at least a little bit unpredictable. That description certainly applies to the collections on display in “An Enduring Vision: Photographs From the Lane Collection,” which runs at the Museum of Fine Arts through March 30, and “Building a Collection: Photography at the Fitchburg Art Museum,” which runs through Sept. 1.

William Lane, one of the MFA collection’s two namesakes, began collecting American Modernist painters and works on paper in the 1950s. He met his future wife, Saundra, in 1962. They began to collect American Modernist photographs, starting with the estate of Charles Sheeler, in 1965.

This addition made perfect sense. These painters and photographers shared numerous concerns: urban life as subject matter, for example, or precision of detail in presentation. Some of the sharing took other forms. One of the painters the Lanes collected, Georgia O’Keeffe, was married to the founding father of American Modernist photography, Alfred Stieglitz. Sheeler was both painter and photographer. His 1915 photograph of a Bucks County, Pa., barn is among the 52 images in the show. In a characteristically witty bit of hanging by curator Karen Haas, it’s next to Édouard-Denis Baldus’s photograph of the Louvre facade, from the 1850s. The juxtaposition of the photographers’ shared frontal approach with the highly divergent building styles and visual textures is striking.

The observant reader will object that American Modernist photographs weren’t being taken in France in the 1850s. And while a picture like Ilse Bing’s “Champ-de-Mars, View From the Eiffel Tower,” from 1931, may not be American, it certainly shares attributes with Sheeler’s work — or that of such contemporaries as, say, Edward Weston (who has seven photographs in the show) or Ansel Adams (who has two) — Baldus’s comes from a very different sensibility no less than a different century.

This is where unpredictability comes in. Photography is as inherently various as the world it records — and any good collector’s taste should be various, too, or at least somewhat. After her husband’s death, in 1995, Saundra Lane began collecting on her own. She broadened the collection to include 19th-century photographs, like the Baldus, and others from later in the 20th century and into the 21st.

AN ENDURING VISION: Photographs From the Lane Collection

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The greatest pleasure “An Enduring Vision” offers is the images themselves, of course: from the delicacy of Irving Penn’s “Gingko Leaves” (one of two color pictures in the show) to the wit of Judy Dater’s “Imogen and Twinka” (part of the wit is in “Imogen” being Imogen Cunningham, one of those classic American Modernists). But there is also the pleasure of seeing how the earlier and later photographs extend and deepen themes and motifs found in the core collection. The subdued magnificence of Joe Deal’s “Butte, Colorado Piedmont” from 2005 is very different from Adams’s views of the American West, but recognizably kin to them.

The Fitchburg Art Museum’s photography collection has 450 images, far fewer than the 6,000 in the Lane Collection. The museum began collection photographs in 1980. And, of course, the FAM collection isn’t the product of just two collectors (though three-quarters of it came from a single donor, Jude Peterson).

What’s striking about the 64 photographs on display is how they manage to suggest coherence without seeming restrictive. Curator Stephen Jareckie has hung the show in roughly chronological order: from an 1863 Alexander Gardner Civil War photograph to several arresting views of Kyoto, Japan, taken by Chester Michalik in 2005.

There are certain emphases — the American Southwest and landscape generally, American Indians, the photographers of the Farm Security Administration — and some very famous images (Dorthea Lange’s “Migrant Mother,” Stieglitz’s “The Steerage”). But there are also three examples of Peter Henry Emerson’s Pictorialism (a school very distant indeed from the FSA aesthetic) and two of Harold Edgerton’s stroboscopic marvels. There are also some real finds, such as Kenro Izu’s platinum-palladium prints of ancient monuments.

Several photographers overlap with those in “An Enduring Vision”: Adams, Weston, Sheeler. Fitchburg, in fact, has on display Sheeler’s camera case. It looks even more nobly battered than that Bucks County barn does. Both shows include an image each from Arno Rafael Minkkinen, whose combining of self-portraiture and landscape manages to be at once an updating of Surrealism and startling commentary on man’s place in the environment. Speaking of combining, the Fitchburg photograph is a gift of Saundra Lane — and one of the Izus was bought with funds donated by her. So besides having personalities of their own, the two collections have an actual personality in common, too.